Vegard's law: a fundamental relation or an approximation?
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K. T. Jacob
Abstract
Vegard's law has been used extensively in mineralogy, metallurgy and materials science for the past six decades. According to the law, unit cell parameters should vary linearly with composition for a continuous substitutional solid solution in which atoms or ions that substitute for each other are randomly distributed. Although the law was postulated on empirical evidence, several cases of both positive and negative deviations from this law have been documented. Its theoretical foundations have not been critically explored. Presented in this communication is an analysis of the law within the framework of solution thermodynamics. It is shown that the deviation from Vegard's law is expected even for thermodynamically ideal solutions when there is a significant difference in lattice parameters of the pure components. The law should be reclassified as an approximation valid for specific conditions. The approximation is valid for ideal solutions when the lattice parameters of the pure components differ by less than 5 %. For solid solutions with positive deviations from ideality, there will always be positive deviations from Vegard's law. For solid solutions with moderately negative deviations from ideality, positive deviation from linearity of lattice parameters caused by size mismatch can be compensated for by the attractive interaction between the components, resulting in compliance with Vegard's law.
References
[1] R.E.Smallman: Modern Physical Metallurgy, Fourth Edition, Butterworths, London (1985) 84.Search in Google Scholar
[2] B.D.Cullity: Elements of X-ray Diffraction, Addison-Wesley, Massachusetts (1967) 352.Search in Google Scholar
[3] W.D.Kingery, H.K.Bowen, D.R.Uhlmann: Introduction to Ceramics, Second Edition, John Wiley, New York (1976) 131.10.1149/1.2133296Search in Google Scholar
[4] L.Vegard: Z. Phys.5 (1921) 17.10.1007/BF01349680Search in Google Scholar
[5] L.Vegard: Z. Cryst.67 (1928) 239.10.1524/zkri.1928.67.1.239Search in Google Scholar
[6] N.Nishiyama, J.Lin, A.Okazaki, M.Iwasaka, K.Hirakawa: Jap. J. Appl. Phys.29 (1990) 369.Search in Google Scholar
[7] E.Kasper, A.Schuh, G.Bauer, B.Holländer, H.Kibbel: J. Cryst. Growth.157 (1995) 68.Search in Google Scholar
[8] C.Y.Fong, W.Weber, J.C.Phillips: Phys. Rev. B14 (1976) 5387.10.1103/PhysRevB.85.099902Search in Google Scholar
© 2007, Carl Hanser Verlag, München
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Articles in the same Issue
- Contents
- Contents
- Editorial
- Computational Thermochemistry
- Gunnar Eriksson 65 years
- Basic
- Vegard's law: a fundamental relation or an approximation?
- Is it a compound or cluster energy formalism?
- Post-optimization elimination of inverted miscibility gaps
- Thermodynamic evaluation of the Au–Sn system
- Applications of thermodynamic calculations to Mg alloy design: Mg–Sn based alloy development
- Thermodynamic modeling of the CoO–SiO2 and CoO–FeO–Fe2O3–SiO2 systems
- Scheil–Gulliver simulation with partial redistribution of fast diffusers and simultaneous solid–solid phase transformations
- Analysis of X-ray extinction due to homogeneously distributed dislocations – Bragg case
- Applied
- Thermodynamic modelling in the ZrO2–La2O3–Y2O3–Al2O3 system
- Thermodynamic optimisation of the FeO–Fe2O3–SiO2 (Fe–O–Si) system with FactSage
- Reassessment of the Al–Mn system and a thermodynamic description of the Al–Mg–Mn system
- Application of FactSage thermodynamic modeling of recycled slags (Al2O3–CaO–FeO–Fe2O3–SiO2–PbO–ZnO) in the treatment of wastes from end-of-life-vehicles
- Bio-inspired syntheses of ZnO-protein composites
- Preparation and characterization of cobalt–bismuth nano- and micro-particles
- Strain rate dependency on deformation texture for pure polycrystalline tantalum
- Notifications
- DGM News
